Existence and continuity results for a nonlinear fractional Langevin equation with a weakly singular source

We study a nonlinear Langevin equation involving a Caputo fractional derivatives of a function with respect to another function in a Banach space. Unlike previous papers, we assume the source function having a singularity. Under a regularity assumption of solution of the problem, we show that the problem can be transformed to a Volterra integral equation with two parameters Mittag-Leffler function in the kernel. Base on the obtained Volterra integral equation, we investigate the existence and uniqueness of the mild solution of the problem. Moreover, we show that the mild solution of the problem is dependent continuously on the inputs: initial data, fractional orders, appropriate function, and friction constant. Meanwhile, a new Henry-Gronwall type inequality is established to prove the main results of the paper. Examples illustrating our results are also presented.